#72223129

order	6400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #12821

Subgroups

	subgroup	G/H
center	C₂	#72223128
commutator	C₂×C₅²×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₄×C₅²×C₈	C₂×C₄
socle	C₂×C₅²	#1802
prime radical	#13105775	C₂²
cosocle	C₂×C₈×C₅⋊D₅	C₂³
hypercenter	C₄×C₈	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#12821
LCS	G ▷ C₂×C₅²×C₈ ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈

1140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^8,5^2|b^2=d,c^2=f,e^2=gh,g^2=h,b^a=be,c^a=cgh,c^b=cf,e^a=eg,e^b=eg,f^a=fh,g^a=gh,g^b=gh,i^a=i^4,i^b=i^3,i^d=i^4,j^a=i^3j,j^b=j^3,j^d=j^4

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7*F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^4);;
gap> SetConjugate(coll, 9, 2, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^3*F.10);;
gap> SetConjugate(coll, 10, 2, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);